Ridit splines with applications to propensity weighting

Given a random variable X, the ridit function RX(·) specifies its distribution. The SSC package wridit can compute ridits (possibly weighted) for a variable. A ridit spline in a variable X is a spline in the ridit RX(X). The SSC package polyspline can be used with wridit to generate an unrestricted ridit-spline basis for an X-variable, with the feature that, in a regression model, the parameters corresponding to the basis variables are equal to mean values of the outcome variable at a list of percentiles of the X-variable. Ridit splines are especially useful in propensity weighting. The user may define a primary propensity score in the usual way, by fitting a regression model of the treatment variable with respect to the confounders, and then using the predicted values of the treatment variable. A secondary propensity score is then defined by regressing the treatment variable with respect to a ridit- spline basis in the primary propensity score. We have found that secondary propensity scores can predict the treatment variable as well as the corresponding primary propensity scores, as measured using the unweighted Somers' D with respect to the treatment variable. However, secondary propensity weights frequently perform better than primary propensity weights at standardizing out the treatment-propensity association, as measured using the propensity- weighted Somers' D with respect to the treatment variable. Also, when we measure the treatment effect, secondary propensity weights may cause considerably less variance inflation than primary propensity weights. This is because the secondary propensity score is less likely to produce extreme propensity weights than the primary propensity score.